這是一個關於微分算子的譜理論研究計畫。這個計畫主要的目的是從微分算子的譜 或者是固有函數的節點去反推非線性微分算子的係數。我們計劃發展制必要數學方法, 以對於譜的反問題和節點反問題取個理論解並對這些辦法發展出數值算法。 我們主要的重點是放在微分方程和邊界條件關於普參數的非線性倚賴性情況。 此 外, 我們都將研究二階微分方程和更高階複雜的非線性微分方程。 我們想要了解譜的反 問題和節點反問題關聯。主要的研究步驟為: - 界定反問題，如何由節點和譜去解反問題; - 研究節點和譜的的解析特性, 漸近行為和結構; 研究固有函數的振盪性質; 。 - 研究唯一性理論; - 提供非線性微分方程反問題的構造性解。 - 建立反問題的可解性的充要條件 - 解的穩定性研究 - 發展數值方法和執行數值驗證。 The proposed project is related to the spectral theory of differential operators. The goal of the project is the investigation of nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics and/or nodal points of root functions. We plan to work out necessary mathematical methods, to obtain solutions for a wide class of inverse spectral and inverse nodal problems and to develop algorithms for their solutions. We will pay the main attention to pencils of differential operators, i.e. to the case of nonlinear dependence of differential equations and boundary conditions on the spectral parameter. Moreover, we will study both second-order differential equations and more complicated higher-order differential equations. We plan to investigate connections between inverse nodal and inverse spectral problems. The main stages of the study are: - to classify the inverse problems, to introduce the nodal and spectral characteristics needed for formulations and solutions of the inverse problems; - to investigate analytic, asymptotic and structural properties of the nodal and spectral characteristics; to study oscillation properties of the root functions; - to obtain uniqueness theorems, to indicate nodal and spectral characteristics which give us information on necessary measurements for the solution of the inverse problems; - to provide constructive procedures for the solutions of inverse nodal and inverse spectral problems for differential equations with nonlinear dependence on the spectral parameter; - to establish necessary and sufficient conditions for the solvability of this class of nonlinear inverse problems, and to describe the corresponding classes of spectral and nodal characteristics; - to study the stability of the solutions of the inverse problems; - to develop numerical methods and to conduct numerical experiments.